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Utilizing an integral representation of smooth functions of d variables proved using properties of delta and Heaviside distributions we estimate variation with respect to half-spaces in terms of \ ows through hyperplanes". Consequently we obtain conditions which guarantee L 2 approximation error rate of order O( 1 p n ) b y one-hidden-layer networks with n sigmoidal perceptrons. KeywordsApproximation of functions, one-hidden-layer sigmoidal networks, estimates of the number of hidden units, variation with respect to half-spaces, integral representation

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a b s t r a c tComplexity of Gaussian-radial-basis-function networks, with varying widths, is investigated. Upper bounds on rates of decrease of approximation errors with increasing number of hidden units are derived. Bounds are in terms of norms measuring smoothness (Bessel and Sobolev norms) multiplied by explicitly given functions a(r, d) of the number of variables d and degree of smoothness r. Estimates are proven using suitable integral representations in the form of networks with continua of hidden units computing scaled Gaussians and translated Bessel potentials. Consequences on tractability of approximation by Gaussian-radial-basis function networks are discussed.

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Quantitative bounds on rates of approximation by linear combinations of Heaviside plane waves are obtained for sufficiently differentiable functions f which vanish rapidly enough at infinity: for d odd and f ∈ C d (R d ), with lower-order partials vanishing at infinity and dth-order partials vanishing as x −(d+1+ε) , ε > 0, on any domain ⊂ R d with unit Lebesgue measure, the L 2 ( )-error in approximating f by a linear combination of n Heaviside plane waves is bounded above by k d f d,1,∞ n −1/2 , where k d ∼ ( d) 1/2 (e/2 ) d/2 and f d,1,∞ is the Sobolev seminorm determined by the largest of the L 1 -norms of the dth-order partials of f on R d . In particular, for d odd and f (x)=exp(− x 2 ), the L 2 ( )-approximation error is at most (2 d) 3/4 n −1/2 and the sup-norm approximation error on R d is at most 68 √ 2(n−1) −1/2 (2 d) 3/4 √ d + 1, n 2.

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